the global dimension of ω-smash coproducts

ISSN 0001-4346, Mathematical Notes, 2013, Vol. 94, No. 4, pp. 499–507. © Pleiades Publishing, Ltd., 2013.Published in Russian in Matematicheskie Zametki, 2013, Vol. 94, No. 4, pp. 541–551.

The Global Dimension of ω-Smash Coproducts*

L. Y. Zhang** and W. PanNanjing Agricultural University, Nanjing, China

Received January 28, 2012

Abstract—We mainly study the global dimension of ω-smash coproducts. We show that if H is aHopf algebra with a bijective antipode SH , and Cω �� H denotes the ω-smash coproduct, then

gl.dim(Cω �� H) ≤ gl.dim(C) + gl.dim(H),

where gl.dim(H) denotes the global dimension of H as a coalgebra.

DOI: 10.1134/S0001434613090204

Keywords: spectral sequence, global dimension, ω-smash coproduct.

1. INTRODUCTIONIn [1], the authors introduced the concepts of R-smash products and ω-smash coproducts. Let A

and B be two algebras, and A �R B = A ⊗ B as vector spaces. The multiplication in A �R B is given bythe formula

mA�B = (mA ⊗ mB)(idA ⊗ R ⊗ idB),

where the map R is denoted by

R : B ⊗ A → A ⊗ B, b ⊗ a �→ Ra ⊗ Rb

and idA denotes the identical map from A to A. If A �R B is an associative algebra with a unit 1A � 1B , itis called an R-smash product. The dual concept of R-smash products was defined by Caenepeel et al.

The global dimension of coalgebras was discussed by several authors. In [2], Doi gave an introductionto the homology theory of comodules over coalgebras and Hopf algebras. In [3], Dascalescu et al. studiedthe global dimension of coalgebras and discussed the class of coalgebras of global dimension less orequal to 1. This class of coalgebras was called exactly hereditary coalgebras. In [4], the authorsproved that if C �α H is a crossed coproduct with a convolution invertible cocycle α, then

gl.dim(C �α H) ≤ gl.dim(C) + gl.dim(H),and showed that the global dimension of a Hopf algebra H as a coalgebra is equal to the injectivedimension of the trivial right H-comodule k. In [5], Lorenz proved that the left global dimension of aHopf algebra H as an algebra is equal the left projective dimension of the trivial H-module k. In [6],Caenepeel et al. proved that

gl.dim(A �R H) ≤ gl.dim(A) + gl.dim(H).

The main contents of this paper are a discussion of the relationship of global dimensions between theω-smash coproduct Cω �� H , the coalgebra C, and the Hopf algebra H as a coalgebra. If H is a Hopfalgebra with a bijective antipode SH , M belongs to MCω��H , the category of right Cω �� H-comodulesand N belongs to Cω��HM, the category of left Cω �� H-comodules, then there exists a third quadrantspectral sequence

Ep,q2 = TorH,p(k,TorC,q(M,N)) =⇒p TorCω��H,n(M,N).

As a consequence, we obtain

gl.dim(Cω �� H) ≤ gl.dim(C) + gl.dim(H).

∗The text was submitted by the authors in English.**E-mail: [email protected]

499

500 ZHANG, PAN

2. PRELIMINARIES

Throughout this paper, we always work over a fixed field k and follow Montgomery’s book [7] forthe terminology concerning coalgebras and comodules, but omit the usual summation indices andsummation symbols.

In [3], the authors gave the following definitions and obtained the following results concerning ω-smash coproducts.

Let C and D be two coalgebras, and consider the linear map

ω : C ⊗ D → D ⊗ C, ω(c ⊗ d) = ωd ⊗ ωc.

Let Cω �� D = C ⊗ D as vector spaces with the comultiplication given by the formula

ΔCω��D = (idC ⊗ ω ⊗ idD)(ΔC ⊗ ΔD)

or

ΔCω��D(c �� d) = (c1 �� ωd1) ⊗ (ωc2 �� d2) (2.1)

for all c ∈ C and d ∈ D. With the notation above, Cω �� D is called an ω-smash coproduct if thecomultiplication ΔCω��D is coassociative, with the counit εCω��D(c �� d) = εC(c)εD(d). Let C and Dbe two coalgebras, and let ω : C ⊗ D → D ⊗ C be a linear map. The map ω is said to be left conormalif

(idD ⊗ εC)ω = εC ⊗ idD, i.e. εC(ωc)ωd = εC(c)d, (2.2)

for all c ∈ C and d ∈ D; ω is said to be right conormal if

(εD ⊗ idC)ω = εD ⊗ idC , i.e., εD(ωd)ωc = εD(d)c, (2.3)

for all c ∈ C and d ∈ D; ω is said to be conormal if ω is left and right conormal.We have the following necessary and sufficient condition for Cω �� D to be an ω-smash coproduct.

Statement 2.1. Let C and D be two coalgebras, and let ω : C ⊗ D → D ⊗ C be a linear map. Thefollowing statements are equivalent:

(1) Cω �� D is an ω-smash coproduct;

(2) the following conditions hold:

(i) ω is conormal;

(ii) the following two pentagonal diagrams are commutative:

C ⊗ Dω ��

idC⊗ΔD

��

D ⊗ CΔD⊗idC �� D ⊗ D ⊗ C

C ⊗ D ⊗ Dω⊗idD

�� D ⊗ C ⊗ D

idD⊗ω

��

(CP1)

C ⊗ Dω ��

ΔC⊗idD

��

D ⊗ CidD⊗ΔC �� D ⊗ C ⊗ C

C ⊗ C ⊗ DidC⊗ω

�� C ⊗ D ⊗ C

ω⊗idC

��

(CP2)

Statement 2.2. Let Cω �� D be an ω-smash coproduct. Then for all c ∈ C and d ∈ D, the maps

p1 : Cω �� D → C, p1(cω �� d) = εD(d)c,p2 : Cω �� D → D, p2(cω �� d) = εC(c)d,

are coalgebra morphisms such that

(p2 ⊗ p1)ΔCω��D = ω(p1 ⊗ p2)ΔCω��D. (2.4)

MATHEMATICAL NOTES Vol. 94 No. 4 2013

THE GLOBAL DIMENSION OF ω-SMASH COPRODUCTS 501

3. SPECTRAL SEQUENCE AND GLOBAL DIMENSION OF ω-SMASH COPRODUCTS

Let C be a coalgebra, H a Hopf algebra with a bijective antipode S, and ω : C ⊗H → H ⊗ C a linearmap such that Cω �� H is an ω-smash coproduct. Define

pC : Cω �� H → C, pC(c �� h) = εH(h)c,pH : Cω �� H → H, pH(c �� h) = εC(c)h,

for all c ∈ C and h ∈ H . For any d = c �� h ∈ Cω �� H , we denote ΔCω��H(d) by d1 ⊗ d2. It is easy toprove that

(pC ⊗ pH)ΔCω��H(c �� h) = (pC ⊗ pH)(c1 �� ωh1 ⊗ ωc2 �� h2)

= εH(ωh1)c1 ⊗ εC(ωc2)h2(2.3)= εH(h1)c1 ⊗ εC(c2)h2 = c ⊗ h,

i.e.,

pC(d1) ⊗ pH(d2) = d. (3.1)

Hence we have

(pH ⊗ pC)ΔCω��H(c �� h)(2.4)= ω(pC ⊗ pH)ΔCω��H(c ⊗ h) = ω(c ⊗ h),

i.e.,

pH(d1) ⊗ pC(d2) = ωpH(d2) ⊗ ωpC(d1). (3.2)

Since ω is conormal, we know that

(pH ⊗ pH)ΔCω��H(c �� h) = (pH ⊗ pH)(c1 ��ω h1 ⊗ω c2 �� h2)

= εC(c1)ωh1 ⊗ εC(ωc2)h2(2.2)= εC(c1)h1 ⊗ εC(c2)h2

= (pH(c �� h))1 ⊗ (pH(c �� h))2,

i.e.,

pH(d1) ⊗ pH(d2) = (pH(d))1 ⊗ (pH(d))2. (3.3)

For the ω-smash coproduct Cω �� H , by MCω��H we denote the category of right Cω �� H-como-dules, and by Cω��HM the category of left Cω �� H-comodules. If M ∈ MCω��H and N ∈ Cω��HM,we use the notation ρM (m) = m(0) ⊗ m(1) and ρN (n) = n(−1) ⊗ n(0) for the right Cω �� H-comodulestructure map and the left Cω �� H-comodule structure map, respectively. Define the cotensor productM �Cω��H N to be the kernel of the map ρM ⊗ idN − idM ⊗ ρN . Since ω is conormal, it is easy tosee that pC : Cω �� H → C is coalgebra epimorphic. Let (M,ρM ) ∈ MCω��H . Then M ∈ MC , whosecomodule structure is given by ρC

M = (idM ⊗ pC)ρM . Indeed, since pC is a coalgebra map, for anyd ∈ Cω �� H , we have

pC(d)1 ⊗ pC(d)2 = pC(d1) ⊗ pC(d2)

and

m(0) ⊗ pC(m(1))1 ⊗ pC(m(1))2 = m(0) ⊗ pC(m(1)) ⊗ pC(m(2)),

i.e.,

(idM ⊗ ΔC)ρCM (m) = (ρC

M ⊗ idC)ρCM (m), M ∈ MC .

Similarly, for any (N, ρN ) ∈ Cω��HM, we have N ∈ CM, where the comodule structure is given byρC

N = (pC ⊗ idN )ρN . From the above comodule structures over C, we can obtain a new cotensor product

M �C N = Ker(ρCM ⊗ idN − idM ⊗ ρC

N ),

induced by the coalgebra epimorphism pC . Let m ⊗ n ∈ M �C N . Then it is easy to see that

m(0) ⊗ pC(m(1)) ⊗ n = m ⊗ pC(n(−1)) ⊗ n(0). (3.4)


502 ZHANG, PAN

By applying ρM ⊗ idC ⊗ ρN to (3.4), we obtain

m(0) ⊗ m(1) ⊗ pC(m(2)) ⊗ n(−1) ⊗ n(0) = m(0) ⊗ m(1) ⊗ pC(n(−2)) ⊗ n(−1) ⊗ n(0). (3.5)

In this section, we assume that ω satisfies the following condition: for any d, d′ ∈ Cω �� H

pC(d1) ⊗ SH(pH(d2))pH(d′) = ωpC(d2) ⊗ SH(pH(d1))ωpH(d′). (3.6)

Lemma 3.1. Let H be a Hopf algebra, and Cω �� H an ω-smash coproduct. Then the inducedcotensor product M �C N is a left H-comodule whose comodule structure is given by

ρ(m ⊗ n) = S(pH(m(1)))pH(n(−1)) ⊗ m(0) ⊗ n(0). (3.7)

Proof. Since

M �C N = Ker(ρCM ⊗ idN − idM ⊗ ρC

N ),

we obtain

H ⊗ (M �C N) = Ker(idH ⊗ ρCM ⊗ idN − idH ⊗ idM ⊗ ρC

N ).

Then

(idH ⊗ ρCM ⊗ idN − idH ⊗ idM ⊗ ρC

N )ρ(m ⊗ n)= SH(pH(m(2)))pH(n(−1)) ⊗ m(0) ⊗ pC(m(1)) ⊗ n(0)

− SH(pH(m(1)))pH(n(−2)) ⊗ m(0) ⊗ pC(n(−1)) ⊗ n(0),

but

SH(pH(m(2)))pH(n(−1)) ⊗ m(0) ⊗ pC(m(1)) ⊗ n(0)

(3.6)= SH(pH(m(1)))

ωpH(n(−1)) ⊗ m(0) ⊗ ωpC(m(2)) ⊗ n(0)

(3.5)= SH(pH(m(1)))

ωpH(n(−1)) ⊗ m(0) ⊗ ωpC(n(−2)) ⊗ n(0)

= SH(pH(m(1)))ωpH(n(−1)2) ⊗ m(0) ⊗ ωpC(n(−1)1) ⊗ n(0)

(3.2)= SH(pH(m(1)))pH(n(−1)1) ⊗ m(0) ⊗ pC(n(−1)2) ⊗ n(0)

= SH(pH(m(1)))pH(n(−2)) ⊗ m(0) ⊗ pC(n(−1)) ⊗ n(0),

so

(idH ⊗ ρCM ⊗ idN − idH ⊗ idM ⊗ ρC

N )ρ(m ⊗ n) = 0,

i.e., ρ(m ⊗ n) ∈ H ⊗ (M �C N), and ρ is well defined.

Let m ⊗ n ∈ M �C N . Then

(idH ⊗ ρ)ρ(m ⊗ n)(3.7)= (idH ⊗ ρ)(SH(pH(m(1)))pH(n(−1)) ⊗ m(0) ⊗ n(0))

= SH(pH(m(2)))pH(n(−2)) ⊗ SH(pH(m(1)))pH(n(−1)) ⊗ m(0) ⊗ n(0)

= SH(pH(m(1)2))pH(n(−1)1) ⊗ SH(pH(m(1)1))pH(n(−1)2) ⊗ m(0) ⊗ n(0)

(3.3)= (SH(pH(m(1))))1(pH(n(−1)))1 ⊗ (SH(pH(m(1))))2(pH(n(−1)))2

⊗ m(0) ⊗ n(0)

= (ΔH ⊗ idM�CN )ρ(m ⊗ n),

(εH ⊗ idM�CN )ρ(m ⊗ n) = εH(SH(pH(m(1))))εH (pH(n(−1)))m(0) ⊗ n(0)

= εCω��H(m(1))m(0) ⊗ εCω��H (n(−1))n(0) = m ⊗ n.

It follows that M �C N is a left H-comodule.



Lemma 3.2. Let H be a Hopf algebra, Cω �� H an ω-smash coproduct. Then, for any M ∈ MCω��H

and any N ∈ Cω��HM, there is a canonical linear isomorphism

k �H (M �C N) ∼= M �Cω��H N,

where k is regarded as the trivial right H-comodule.

Proof. Let ϕ : k ⊗M ⊗N → M ⊗N be the natural isomorphism and ϕ0 : k ⊗ (M �C N) → M �C Nthe restriction map. In what follows, we shall show that

ϕ0(k �H (M �C N)) = M �Cω��H N.

Let us take m ⊗ n ∈ M �Cω��H N , i.e., m(0) ⊗ m(1) ⊗ n = m ⊗ n(−1) ⊗ n(0). Then

m(0) ⊗ m(1) ⊗ m(2) ⊗ n = m(0) ⊗ m(1) ⊗ n(−1)⊗(0),

and so

SH(pH(m(1)))pH(m(2)) ⊗ m(0) ⊗ n = SH(pH(m(1)))pH(n(−1)) ⊗ m(0) ⊗ n(0),

i.e., 1H ⊗ m ⊗ n = SH(pH(m(1)))pH(n(−1)) ⊗ m(0) ⊗ n(0). So

1H ⊗ m ⊗ n = ϕ−10 (m ⊗ n) ∈ k �H (M �C N),

i.e.,

M �Cω��H N ⊆ ϕ0(k �H (M �C N)).

Conversely, let us now take a nonzero element α ⊗ m ⊗ n ∈ k �H (M �C N), i.e.,

α ⊗ m ⊗ n ∈ Ker(ρk ⊗ idM�CN − idk ⊗ ρ),

so

α ⊗ 1H ⊗ m ⊗ n = α ⊗ SH(pH(m(1)))pH(n(−1)) ⊗ m(0) ⊗ n(0)

in k ⊗ H ⊗ M ⊗ N . Then

1H ⊗ m ⊗ n = SH(pH(m(1)))pH(n(−1)) ⊗ m(0) ⊗ n(0)

in H ⊗ M ⊗ N . This shows that

1H ⊗ m(0) ⊗ m(1) ⊗ n = SH(pH(m(2)))pH(n(−1)) ⊗ m(0) ⊗ m(1) ⊗ n(0). (3.8)

By applying idH ⊗ idM ⊗ PH ⊗ idN to (3.8), we obtain

pH(m(1)) ⊗ m(0) ⊗ n = pH(m(1))SH(pH(m(2)))pH(n(−1)) ⊗ m(0) ⊗ n(0)

or

pH(m(1)) ⊗ m(0) ⊗ n = pH(n(−1)) ⊗ m ⊗ n(0). (3.9)

The above equality (3.9) implies

m(0) ⊗ pC(m(1)) ⊗ pH(m(2)) ⊗ n = m(0) ⊗ pC(m(1)) ⊗ pH(n(−1)) ⊗ n(0).

According to (3.1), we can write

m(0) ⊗ m(1) ⊗ n = m(0) ⊗ pC(m(1)) ⊗ pH(n(−1)) ⊗ n(0). (3.10)

On the other hand, since m ⊗ n ∈ M �C N , applying idM ⊗ idC ⊗ ρN to (3.4), we obtain

m(0) ⊗ pC(m(1)) ⊗ n(−1) ⊗ n(0) = m ⊗ pC(n(−2)) ⊗ n(−1) ⊗ n(0).

So

m(0) ⊗ pC(m(1)) ⊗ pH(n(−1)) ⊗ n(0) = m ⊗ pC(n(−2)) ⊗ pH(n(−1)) ⊗ n(0),

and hence we obtain

m(0) ⊗ pC(m(1)) ⊗ pH(n(−1)) ⊗ n(0) = m ⊗ n(−1) ⊗ n(0). (3.11)


504 ZHANG, PAN

By (3.10) and (3.11), we know that

m(0) ⊗ m(1) ⊗ n = m ⊗ n(−1) ⊗ n(0),

i.e., m ⊗ n ∈ M �Cω��H N . Thus,

ϕ0(k �H (M �C N)) ⊆ M �Cω��H N ,

which completes the proof.

Statement 3.3. Let H be a Hopf algebra, Cω �� H an ω-smash coproduct, and M ∈ MCω��H . Thenthere exists an isomorphism of left H-comodules:

H ⊗ M ∼= M �C (Cω �� H),

where the H-coaction of H ⊗ M is via the comultiplication ΔH in H .

Proof. Define the linear maps

F : M �C (Cω �� H) → H ⊗ M, F (m ⊗ d) = SH(pH(m(1)))pH(d) ⊗ m(0)

and

G : H ⊗ M → M �C (Cω �� H), G(h ⊗ m) = m(0) ⊗ (m(1)C ⊗ m(1)

Hh),

where we use the following notation for the Cω �� H-comodule of M :

ρM (m) = m(0) ⊗ m(1) ≡ m(0) ⊗ mC(1) ⊗ mH

(1).

The left C-comodule structure of Cω �� H is given by

ρCCω��H = (pC ⊗ idCω��H)ΔCω��H .

First, F and G are well defined. In fact, for any m ∈ M and h ∈ H , we have

(ρCM ⊗ idCω��H − idM ⊗ ρC

Cω��H )(G(h ⊗ m))

= (ρCM ⊗ idCω��H − idM ⊗ ρC

Cω��H )(m(0) ⊗ (m(1)C ⊗ m(1)

Hh))

= (m(0) ⊗ pC(m(1)) ⊗ (mC(2) ⊗ mH

(2)h)) − (m(0) ⊗ pC((mC(1) ⊗ mH

(1)h)1) ⊗ (mC(1) ⊗ mH

(1)h)2)

= (m(0) ⊗ mC(1)εH(mH

(1)) ⊗ (mC(2) ⊗ mH

(2)h))

− (m(0) ⊗ (mC(1))1 ⊗ εH(ω(mH

(1)h)1)ω(mC(1))2 ⊗ (mH

(1)h)2)

(2.3)= (m(0) ⊗ mC

(1) ⊗ (mC(2) ⊗ εH(mH

(1))mH(2)h))

− (m(0) ⊗ (mC(1))1 ⊗ (mC

(1))2 ⊗ εH((mH(1)h)1)(mH

(1)h)2)

= (m(0) ⊗ mC(1) ⊗ (mC

(2) ⊗ mH(1)h)) − (m(0) ⊗ mC

(1) ⊗ (mC(2) ⊗ mH

(1)h)) = 0,

so G(h ⊗ m) ∈ M �C (Cω �� H), where we have used the fact that ω is conormal.Second, for any h ∈ H and m ∈ M , and d ∈ Cω �� H , we have

FG(h ⊗ m) = F (m(0) ⊗ mC(1) ⊗ mH

(1)h) = SH(pH(m(1)))pH(mC(2) ⊗ mH

(2)h) ⊗ m(0)

= SH(pH(m(1)))εC(mC(2))m

H(2)h ⊗ m(0) = SH(pH(m(1)))pH(m(2))h ⊗ m(0)

(3.3)= SH(pH(m(1))1)pH(m(1))2h ⊗ m(0) = h ⊗ m,

GF (m ⊗ d) = G(SH(pH(m(1)))pH(d) ⊗ m(0)) = m(0) ⊗ (mC(1) ⊗ mH

(1)SH(pH(m(2)))pH(d))

= m(0) ⊗ mC(1) ⊗ mH

(1)SH(εC(mC(2))m

H(2))pH(d) = m(0) ⊗ mC

(1)εH(mH(1)) ⊗ pH(d)

= m(0) ⊗ pC(m(1)) ⊗ pH(d) = m(0) ⊗ pC(d1) ⊗ pH(d2)(3.1)= m ⊗ d,



where we have used the fact that

m(0) ⊗ pC(m(1)) ⊗ d = m ⊗ pC(d1) ⊗ d2,

since m ⊗ d ∈ M �C (Cω �� H). Thus, F and G are inverse to each other.Finally, we can show that the map F is H-colinear, because

(idH ⊗ F )ρ(m ⊗ d)(3.7)= (idH ⊗ F )(SH(pH(m(1)))pH(d1) ⊗ m(0) ⊗ d2)

= SH(pH(m(1)))pH(d1) ⊗ SH(pH(m(0)(1)))pH(d2) ⊗ m(0)(0)

= SH(pH(m(2)))pH(d1) ⊗ SH(pH(m(1)))pH(d2) ⊗ m(0)

(3.3)= (SH(pH(m(1)))pH(d))1 ⊗ (SH(pH(m(1)))pH(d))2 ⊗ m(0)

= (ΔH ⊗ idM )F (m ⊗ d),

so F is a left H-comodule map, where ρ is the left H-comodule structure map of M �C (Cω �� H).

Given a coalgebra C, the cotensor product is a bifunctor · �C · : MC × CM → kM in the categoryof k-spaces, and is also left exact. Given M ∈ MC and N ∈ CM, we consider the functors

M �C · : CM → kM and · �CN : MC → kM,

and their right derived functors Rn(M �C · ) and Rn( · �C N). According to [2, p. 32], a comodule isinjective if and only if it is coflat. If M and N are injective, then they are coflat, thus M �C · and · �C Nare exact by [2]. From [3], we have

Rn(M �C · )(N) = Rn( · �C N)(M),

which is denoted by TorC,n(M,N). So, for a right C-comodule M , inj.dim(M) ≤ n if and only ifTorC,n+1(M,N) = 0 for any left C-comodule N , and for a left C-comodule N , inj.dim(N) ≤ n if andonly if TorC,n+1(M,N) = 0 for any right C-comodule M . For the right derived functor of the cotensorproduct, we have a third spectral sequence over Cω �� H .

Theorem 3.4. Let H be a Hopf algebra and Cω �� H an ω-smash coproduct. Then, for anyM ∈ MCω��H and N ∈ Cω��HM, there exists a third quadrant spectral sequence:

Ep,q2 = TorH,p(k,TorC,q(M,N)) =⇒p TorCω��H,n(M,N),

where n = p + q.

Proof. By Lemma 3.1, M �C N ∈ HM. Define the functors

G = M �C · : Cω��HM → HM, G(N) = M �C N,

F = k �H · : HM → kM, F (U) = k �H U.

Then F is left exact as a cotensor product.By Lemma 3.2, we have

FG(N) = k �H (M �C N) ∼= M �Cω��H N,

so FG ∼= M �Cω��H · and Rn(FG) = TorCω��H,n(M, · ) for right derived functors.If N ∈ Cω��HM is injective, then G(N) is right F-acyclic, i.e., for any p ≥ 1, (RpF )(G(N)) = 0.

Indeed, if taking N = Cω �� H , by Proposition 3.3, G(N) = M �C (Cω �� H) ∼= H ⊗ M , which isa free H-comodule. Since every free comodule is injective, G(N) is a injective H-comodule, then,(RpF )G(N) = 0, i.e., G(N) is right F-acyclic.

Applying a classical result from [8, Theorem 11.38], we obtain the following third quadrant spectralsequence

Ep,q2 = TorH,p(k,TorC,q(M,N)) =⇒p TorCω��H,n(M,N),

which complete our proof.


506 ZHANG, PAN

The characterization of the injective dimension of comodule in terms of the functors Torn fromTheorem 3.4 or Theorem 3.1 in [4] now shows the following.

Statement 3.5. If Cω �� H is an ω-smash coproduct, then

gl.dim(Cω �� H) ≤ gl.dim(C) + gl.dim(H).

Example 3.6. On the polynomial ring k[x]/(x2), we introduce a coalgebra structure as follows: assumeC = k[x]/(x2) with comultiplication ΔC and counit εC :

ΔC : k[x]/(x2) → k[x]/(x2) ⊗ k[x]/(x2),ΔC(x) = x ⊗ 1C + 1C ⊗ x, ΔC(1C) = 1C ⊗ 1C ,

εC : k[x]/(x2) → k, εC(x) = 0, εC(1C) = 1.

Let H = kZ2 be a group algebra (Hopf algebra), where Z2 = {1H , g} is a group with unit 1H . Definea map ω : C ⊗ H → H ⊗ C as follows:

1C ⊗ 1H �→ 1H ⊗ 1C , 1C ⊗ g �→ g ⊗ 1C ,

x ⊗ 1H �→ g ⊗ x, x ⊗ g �→ 1H ⊗ x.

Then we have an ω-smash coproduct Cω �� H such that condition (3.6) holds.As a matter of fact, by Proposition (2.1), it is easy to show that Cω �� H is an ω-smash coproduct.

Since εC(x) = 0, if d′ = x ⊗ 1H or d′ = x ⊗ g, we know that PH(d′) = 0 and

pC(d1) ⊗ SH(pH(d2))pH(d′) = 0 = ωpC(d2) ⊗ SH(pH(d1))ωpH(d′).

SinceΔCω��H(x ⊗ g) = (idC ⊗ ω ⊗ idH)(ΔC ⊗ ΔH)(x ⊗ g)

= (idC ⊗ ω ⊗ idH)((x ⊗ 1C + 1C ⊗ x) ⊗ (g ⊗ g))= x ⊗ g ⊗ 1C ⊗ g + 1C ⊗ 1H ⊗ x ⊗ g,

by taking d = x ⊗ g and d′ = 1C ⊗ g, we can write

pC(d1) ⊗ SH(pH(d2))pH(d′) = pC(x ⊗ g) ⊗ SH(pH(1C ⊗ g))pH (1C ⊗ g)+ pC(1C ⊗ 1H) ⊗ SH(pH(x ⊗ g))pH(1C ⊗ g)

= x ⊗ g2 + 0 = x ⊗ 1H ,ωpC(d2) ⊗ SH(pH(d1))ωpH(d′) = ωpC(1C ⊗ g) ⊗ SH(pH(x ⊗ g))ωpH(1C ⊗ g)

+ ωpC(x ⊗ g) ⊗ SH(pH(1C ⊗ 1H))ωpH(1C ⊗ g)= 0 + x ⊗ 1H = x ⊗ 1H .

So

pC(d1) ⊗ SH(pH(d2))pH(d′) = ωpC(d2) ⊗ SH(pH(d1))ωpH(d′).

The other cases can be proved similarly. Condition (3.6) is satisfied.In this case, we obtain the inequality

gl.dim(k[x]/(x2)ω ��kZ2) ≤ gl.dim(k[x]/(x2)) + gl.dim(kZ2).

If char k = 0, then, by Theorem 7.4.6 (the Larson–Radford Theorem) from [9], we know that kZ2 isa cosemisimple Hopf algebra, so gl.dim(kZ2) = 0. Hence we have

gl.dim(k[x]/(x2)ω ��kZ2) ≤ gl.dim(k[x]/(x2)).

Corollary 3.7. Let H be a cosemisimple Hopf algebra, and Cω �� H an ω-smash coproduct. Then

(1) if C is cosemisimple, then Cω �� H is also cosemisimple;

(2) if C is hereditary, then Cω �� H is also hereditary.



Proof. As is well known, gl.dim(C) = 0 if C is cosemisimple, and gl.dim(C) ≤ 1 if C is hereditary, sothis conclusion holds by Proposition 3.5.

The following lemma is Proposition 3.6 from [4].

Lemma 3.8. Let C be a cocommutative coalgebra such that C∗ is Noetherian. Then

gl.dim(C) = gl.dimalg(C∗).

By Proposition 3.5 and Lemma 3.8, we obtain

Corollary 3.9. Let C be a coalgebra, H a cocommutative Hopf algebra such that H∗ is Noetherianas an algebra. If Cω �� H is an ω-smash coproduct, then

gl.dim(Cω �� H) ≤ gl.dim(C) + gl.dimalg(H∗).

ACKNOWLEDGMENTS

The author Liang-yun Zhang would like to thank Professor Yong-chang Zhu for inviting him to visitHong Kong University of Science and Technology.

This work was supported by the Research Fund for the Doctoral Program of Higher Education ofChina (grant no. 20100097110040) and the Fundamental Research Funds for the Central Universities(grant no. KYZ201125).

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